CAREER: Optimal Transport and Dynamics in Machine Learning

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). The goal of machine learning is to develop algorithms that find meaningful patterns in data. Initially, such algorithms led to breakthroughs in our digital lives, from automated language translation to improved online search. Increasingly, they impact every aspect of life, from medical image analysis to fraud detection. Machine learning has also risen to paramount importance in the sciences, as researchers use the same algorithms to analyze datasets, test hypotheses, and make predictions, accelerating the pace of scientific discovery. However, despite its success, many foundational questions of machine learning remain poorly understood: What is behind the surprising success of neural networks and when might they fail? How can algorithms be tailored to individual scientific experiments, to leverage centuries of domain specific knowledge as they extend the reach of an analysis? To answer these questions, the investigator will study the mathematical foundations of machine learning, using tools from optimal transport and partial differential equations. This research will be integrated with educational opportunities for both undergraduate and graduate students. The investigator will hold undergraduate research symposia to improve awareness of campus research opportunities, with the goal of increasing the number of students conducting research projects in applied mathematics. The investigator will also develop a new graduate course on optimal transport and machine learning, the lectures from which will be made publicly available, and organize an early-career researcher workshop geared to graduate students in the western United States, which will provide students in applied mathematics with an opportunity to learn from a cadre of well-established researchers, as well as to present their own work. At the heart of this research program are three main projects. In the first project, the investigator will study the role of nonlocal interactions in the training dynamics of two-layer neural networks, analyzing how the interplay between model selection, data distribution, and regularity affects robustness and rate of convergence to optimum. In the second project, the investigator will develop particle methods for sampling and control theory based on nonlinear diffusions. In the third project, the investigator will use new optimal transport metrics to develop interpretable machine learning methods for analyzing data from multiple components of a scientific experiment. These methods will then be applied to machine learning tasks in particle physics, including classification of events at the Large Hadron Collider. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.